"To accommodate the screws you'd want to anchor it tightly through the floor, notch the tile before you install them so you don't later have to try to drill through the tile. This is fairly easy to do on the wetsaw or with a grinder and the proper cup. Just measure carefully and ensure you get good coverage with the thinset under that edge and it will be fine."
What I was gonna say.....
Just did that very thing on a job, didnt take a picture though. Much easier than drilling through the tile. Just line up the flange in the way that it will be fixed when the toilet goes on and you can figure out where to make the notches pretty easily.
The flange on my last job also had some movement, It wasnt really the flange moving, more that i was lifting the pipe in the joist space slightly. (ABS)
I wish more bathroom reno's I worked on would change the flange, around here they typicaly leave it, tile around it, and then add spacers